The question asks to sketch the subset of $\{z\ \epsilon\ C : |Z-1|+|Z+1|=4\}$
Here is my working:
$z=x+yi$
$|x+yi-1| + |x+yi+1|=4$
$\sqrt{ {(x-1)}^2 + y^2} + \sqrt{{(x+1)}^2+y^2}=4$
${ {(x-1)}^2 + y^2} + {{(x+1)}^2+y^2}=16$
$x^2 – 2x+1+y^2+x^2+2x+1+y^2=16$
$2x^2+2y^2+2=16$
$x^2+y^2=7$
$(x-0)^2+(y-0)^2=\sqrt7$
=This is a circle with center $0$ and radius $\sqrt7$
My answer is different from the correct answer given: "This is an ellipse with foci at $-1$ and $1$ passing through $2$"
I have no idea how to get to this answer. Could someone please help me here?
Best Answer
Or in another but similar parametrization to Yves':
\begin{align} \sqrt{a+b}+\sqrt{a-b}&=4\\ \text{after squaring: }a+\sqrt{a^2-b^2}&=8\\ \text{rearrange and square: }a^2-b^2&=(8-a)^2=64-16a+a^2\\ 16a-b^2&=64 \end{align} where $a=x^2+y^2+1$ and $b=2x$.